Problem 47 - A Complete Resource Book in Mathematics for JEE Main

Question

Suppose a population \(A\) has 100 observations 101 , \(102, \ldots, 200\), and another population \(B\) has 100 observations \(151,152, \ldots, 250\). If \(V_{A}\) and \(V_{B}\) represent the variances of the two populations, respectively, then \(\frac{V_{A}}{V_{B}}\) is (A) 1 (B) \(9 / 4\) (C) \(4 / 9\) (D) \(2 / 3\)

Step-by-Step Solution

Answer

\(\frac{V_A}{V_B} = 1\) (Answer is (A) 1)

Step 1: Calculate the Mean of Population A

The observations in population A are 101, 102, ..., 200. This creates an arithmetic sequence with 100 terms, where the first term \( a_1 = 101 \) and the last term \( a_n = 200 \). The mean \( \mu_A \) is calculated using the formula for the mean of an arithmetic sequence: \( \mu_A = \frac{{a_1 + a_n}}{2} \). Substitute the given values to find \( \mu_A = \frac{{101 + 200}}{2} = 150.5 \).

Step 2: Calculate the Variance of Population A

Variance \( V_A \) is given by \( V_A = \frac{1}{n} \sum_{i=1}^n (x_i - \mu_A)^2 \). Since the sequence is arithmetic with common difference \(d = 1\), the variance formula for an arithmetic sequence can simplify to \( V_A = \frac{(n^2 - 1)}{12} \), where \( n = 100 \) is the number of observations. Thus, \( V_A = \frac{(100^2 - 1)}{12} = \frac{9999}{12} = 833.25 \).

Step 3: Calculate the Mean of Population B

The observations in population B are 151, 152, ..., 250. As before, this is an arithmetic sequence with 100 terms, where the first term is 151 and the last term is 250. The mean \( \mu_B \) is calculated as \( \mu_B = \frac{{151+250}}{2} = 200.5 \).

Step 4: Calculate the Variance of Population B

Using the same process for calculating variance of an arithmetic sequence as in Step 2, \( V_B = \frac{(100^2 - 1)}{12} = 833.25 \). This is identical to \( V_A \) because the length of the intervals \( (d = 1) \) and the number of terms in the sequence are the same.

Step 5: Calculate the Ratio of Variances

Now, calculate \( \frac{V_A}{V_B} = \frac{833.25}{833.25} = 1 \). Hence, the ratio of the variances of populations \( A \) and \( B \) is 1.

Key Concepts

Understanding Arithmetic Sequences

An arithmetic sequence is a list of numbers with a definite pattern. The pattern involves each number in the sequence being generated by adding a fixed, constant number called the "common difference" to the previous number. For instance, if you start with 101 and keep adding 1, you get a series like 101, 102, 103, and so on. Such sequences are prevalent in mathematical problems and have unique properties.

• First term: The initial number in the sequence.
• Common difference: The consistent interval added to each term.
• Arithmetic sequence formula: If the first term is denoted by \( a_1 \), and the \( n \)-th term by \( a_n \), then \( a_n = a_1 + (n-1) imes d \), where \( d \) is the common difference.

This understanding helps in spotting patterns and simplifying calculations such as finding means and variances in further steps.

Mean Calculation in Arithmetic Sequences

The mean of an arithmetic sequence is important as it gives us the average value of all its terms. This average is crucial for many statistical computations such as variance. The formula to calculate the mean of an arithmetic sequence is:
\[ \mu = \frac{a_1 + a_n}{2} \]

• \( a_1 \): First term of the sequence.
• \( a_n \): Last term of the sequence.

For example, the mean for population A, which consists of numbers from 101 to 200, is calculated by adding the first and the last numbers and dividing by 2, yielding a mean of 150.5.
This method simplifies mean calculation in cases involving arithmetic sequences.

Understanding Population Variance

Population variance is a measure of how much individual values in a data set differ from the mean of the data set. In simpler terms, it tells us the spread or the distribution of the data. For arithmetic sequences, there's a special formula for calculating variance due to the uniform intervals between terms.
The variance \( V \) can be calculated using the formula:
\[ V = \frac{1}{n} \sum_{i=1}^n (x_i - \mu)^2 \]For arithmetic sequences, this simplifies to:
\[ V = \frac{(n^2 - 1)}{12} \] where \( n \) is the number of terms.
Population variance, like in this scenario, is particularly important for understanding data variation.

Exploring Variance Ratio

Variance ratio is a comparison of variance measures across different data sets. It gives us an understanding of differences in data spread between two or more populations. For populations A and B in this exercise, the variance ratio compares the variance of A to that of B.
To find the variance ratio \( \frac{V_A}{V_B} \), we observe the variances \( V_A \) and \( V_B \) were calculated as 833.25 each. This makes their ratio:
\[ \frac{V_A}{V_B} = \frac{833.25}{833.25} = 1 \]
This indicates equal variability or spread in both populations. Understanding the variance ratio helps assess relative consistency or disparity between data sets.